The last five years has seen the emergence of the new field of quantum information processing. The behaviour of individual microscopic particles—atoms, electrons and photons etc., which is described by the theory of quantum mechanics, is completely different from the behaviour of macroscopic objects made of large numbers of such particles. The key realisation which led to the emergence of the field is that quantum mechanics allows the processing of information in fundamentally new ways. This offers the possibility of immense advantages over current information processing technologies which are all essentially based on classical mechanics.
Specific tasks for which it has been proven that quantum information processing offers advantages over classical information processing include:                Communication. In certain circumstances quantum communication is more powerful than classical communication. For instance the amount of quantum communication required to perform certain computational tasks can be significantly smaller than the amount of classical communication. (This aspect is known as communication complexity.) Quantum communication also allows novel, and in principle unbreakable, cryptographic schemes.        Computation. A computer that runs according to the rules of quantum mechanics can perform certain computations faster than a classical computer. In particular there has been described a quantum algorithm for factorising integers which is exponentially faster than any known classical algorithm.        
Just as in classical information processing, errors occur in quantum information processing, and these have to be corrected. However quantum errors are fundamentally different in character from classical errors, and the correction of quantum errors requires new techniques.
When quantum communication was first proposed it was felt that interactions of the system with the environment, and the consequent loss of information into the environment, would produce errors in communication which would be un-correctable even in principle (i.e. it was felt that quantum error-correction would be ruled out by the laws of quantum mechanics). However it was shown in 1995-96, using two different approaches, that quantum error correction was in fact possible.
On the one hand it was shown in C H Bennett, et al, Phys Rev Lett 76 (1996) 722, that entanglement could be protected against decoherence with the environment. The basic idea of this scheme known as entanglement purification, is that from several impure entangled states one can obtain, using only local operations and classical communication, a smaller number of states with higher entanglement. These entangled states can then be used in communication protocols, for example teleportation.
On the other hand it was shown by P W Shor, Method for reducing decoherence in a quantum computer memory, U.S. Pat. No. 5,768,297; P W Shor, Phys Rev A, 52 (1995) R2493 and A M Steane, Phys Rev Lett, 77 (1996) 793, that a quantum memory could be protected against interactions with the environment. The basic idea in this case is to encode each logical quantum bit (qubit) in an entangled superposition of several qubits. The encoded state is chosen in such a way that if a small number of errors affect the individual qubits in the superposition, the errors do not destroy the encoded qubit.
The connections between these two results were studied in C H Bennett, D P DiVincenzo, J A Smolin, W K Wootters, Phys Rev, A54 (1996) 3824-3851, by exploiting the fact that a memory can be viewed as a communication scheme in which the communication takes place in time rather than in space. In particular it was shown in C H Bennett, D P DiVincenzo, J A Smolin, W K Wootters, Phys Rev, A54 (1996) 3824-3851, that a protocol for purification of entanglement that uses only one way classical communication can be mapped into a protocol for the protection of a quantum memory against decoherence. Shor-Steane codes can also be used for error correction for communication by first encoding the qubits using Shor-Steane codes, sending the qubits through the noisy channel and then decoding.
These results subsequently led to the proof that one could in principle realise fault tolerant quantum computation as described in P W Shor, Proc 37th Symposium on the Foundations of Computer Science (Los Alamitos, Calif.; IEEE Comp Sci Press, (1996) p 15.
In all the above work the authors were not concerned with technological feasibility, but rather with proving a point of principle. These protocols are relevant for multi-particle quantum systems and for errors which act locally on the individual particles. The primary difficulty with all these protocols is that in order to be implemented they require controlled interactions between many particles. This is technically impossible at present and is likely to remain so in the foreseeable future.
Braunstein in Nature 394 (1998), 47 and Lloyd and Slotine in Phys Rev Letters 80 (1998), 4088, have proposed methods for implementing Shor-Steane error correction protocols without multi-particle interactions. Their methods are based on the use of continuous quantum variables for quantum logic (to be opposed to the discrete variables, for instance qubits, used in earlier work). The quadratures of the electromagnetic field provide a convenient realisation of such variables. It was shown in Phys Rev. Letters 90 (1998) 4084, that quantum error correction codes could be generalised to continuous quantum variables. These methods apply to a particular physical system, namely light, and require the use of a special type of quantum states: squeezed states of light. Highly squeezed states, which are needed for the Braunstein and Lloyd and Slotine protocols are technologically difficult to prepare and maintain. Furthermore these protocols use active detection of errors and active error correction.
Bouwmeester in quant-ph/006108 has an alternative communication protocol. His method relies on distributing a standard entangled state (a singlet) between two parties. The protocol is limited to correcting classical-type commuting errors (such as spin-flips) when the state propagates through a noisy channel. It does not correct the arbitrary interactions with the environment which occur in the framework used by Bouwmeester. Furthermore the Bouwmeester protocol avoids multi-particle interactions at the expense of using sources which emit very special states (of GHZ-type). Even in the simplest case of correction of a single spin-flip on one party of a singlet propagating through a noisy channel, such sources are at the current limit of technology. Correction of more spin-flip errors requires sources which are beyond the limits of current technology.
It is an aim of the present invention to provide a method and system for handling information which will remove errors introduced by noise. The aim is to provide a method and system which is widely applicable not only to quantum information but also to classical wave signals.
According to one aspect of the present invention there is provided a method of handling information in a noisy environment in the form of at least one information carrying mode, comprising the steps of generating a plurality of encoded modes by linear transformation of the information carrying mode, said encoded modes being provided on respective independent channels; linearly combining the encoded modes to generate at least one decoded mode; providing a set of receiver channels for receiving the decoded mode wherein one of said set of receiver channels is designated as a useful channel; and supplying as a useful signal the decoded mode if it is received on the useful channel and discarding the decoded mode if it is received on the other receiver channels of the set, wherein the useful signal represents said information substantially uncorrupted by noise.
According to another aspect of the invention there is provided a system for handling information in a noisy environment in the form of at least one information carrying mode, the system comprising: means for generating a plurality of encoded modes by linear transformation of the information carrying mode, said encoded modes being provided on respective independent channels; means for linearly combining the encoded modes to generate at least one decoded mode; a set of receiver channels for receiving the at least one decoded mode wherein one of said set of receiver channels is designated as a useful channel; and means for supplying as a useful signal the decoded mode if it is received on the useful channel and discarding the decoded mode if it is received on the other receiver channels of the set, wherein the useful signal represents said information substantially uncorrupted by noise.
The word “mode” used herein applies equally well to systems described by classical or quantum mechanics. In the quantum case it is used, according to its standard meaning in the second quantized description of quantum systems, to denote a set of states, including states with zero, one, two or more particles. In quantum mechanics, one can get super-positions of such states. Thus a first information carrying mode can provide a set of states of no particles in mode 1, one particle in mode 1, etc. In quantum mechanics, a mode can be populated by less than one particle in the sense that the state of the mode can be a superposition of the state with no particles and the state of one particle, for example. In the case of classical wave systems, the word “mode” refers to a basic solution of the appropriate wave equation.
In describing our inventions, we make use of two standard ways of treating quantum systems: the first and second quantization languages. The language of second quantization is more general and can be used in all situations for systems containing any number of particles. Hence all the quantum protocols in this document could have been described in the second quantized language (using the notions of quantum modes, creation and annihilation operators etc.). The first quantization language is, however, more convenient and simpler for systems containing a fixed number of particles. In this document both languages are used, according to which one seems simplest in the given situation.
A key element of our protocols is that the encoding and decoding operations are realised by “linear” transformations. In the case of quantum systems, by linear we mean that the effective interaction implementing the encoding and decoding, as it applies to the space of states we are interested in, is linear; i.e. its effect on the creation operators associated to the quantum modes (in the second quantized language) is such that they undergo linear transformations (for other states, outside the space of states in which we are interested, the true interaction may be non-linear, but this is irrelevant for our protocols). In particular this means that all our single particle and entanglement distribution protocols are linear, by definition. In the context of classical waves, the equivalent notion is that the classical modes undergo linear transformation.
Where an information carrying mode is populated by at least one particle, the encoded modes can be thought of as a plurality of states of the particle which, once generated, can be linearly combined to reconstitute the at least one particle. When one particle is reconstituted, it is detectable at only one of the receiver channels, which may or may not be the useful channel. In that scenario, the useful signal is constituted by particles which are detected on the useful channel, while particles detected on the other receiver channels are discarded.
The term “useful channel” is used to denote those of the receiver channels at which any particle which arrives is in a good, substantially noise free state. As explained later, the underlying theory indicates that particles which arrive on the useful channel will constitute a useful, substantially noise free signal. Some particles will not arrive at the useful channel. The probability of particles appearing in the useful channel is linked to the noise level. Thus, by using the technique of the present invention it is possible to utilise the signal which appears at the useful channel on the basis that it is substantially noise free. Conversely, particles/signals appearing on the other channels are completely ignored or “discarded”. That is, those particles are not used in the output signal or used to modify it.
By the phrase “substantially uncorrupted by noise”, it is meant that the signal contains substantially the same information as in its original state. The invention is particularly suitable for the removal of noise in cases where the noise on each channel is independent of that on any other channel.
It is indicated above that the useful signal “represents” the original information substantially uncorrupted by noise. That is, the information can be recovered in a form substantially identical to its original form, but with reduced amplitude/power (representing the fact that a part of the original signal has been discarded). However it is also possible for the signal which is received to be a modified version of the original information, by virtue of the fact that the channels have been subject to a linear or logical manipulation between the encoding and decoding.
It will be appreciated that the invention is applicable to communication and/or storage as discussed in more detail in the following.
The invention has several important facets.
No multi-particle interactions are needed to provide a substantially noise free particle or signal. This is particularly important since at the present time controlled inter-particle interaction is technologically unfeasible.
Moreover, the encoding and combining steps can be carried out by passive transformations. It had earlier not been thought possible to eliminate noise using only passive transformations.
According to another aspect of the present invention there is provided a communication system for transmitting information in a noisy environment comprising: an input for receiving a physical information signal in the form of a wave; a splitter for splitting the signal into a plurality of transmission components; a plurality of transmission paths, each carrying a respective transmission component; a combiner for combining the transmission components and generating a set of output signals on respective output channels, wherein one of said output channels is designated as a useful channel; a detector for detecting the output signal on the useful channel, wherein the output signals on the other output channels are discarded and wherein the output signal on the useful channel contains said information substantially uncorrupted by noise.
A further aspect provides an optical communications system for transmitting information in a noisy environment, comprising: an input for receiving an optical information signal; a splitter for splitting the optical information signal into a plurality of optical beams; a combiner for combining the optical beams and for generating a set of separated optical output signals, one of said signals being designated as a useful signal; and a detector located to pick up the useful output signal, wherein the other output signal are discarded.
The invention can also be used to generate particle states correlated to a predetermined degree. According to this aspect there is provided a method of generating particle states correlated to a predetermined degree at separated locations in a noisy environment, the method comprising: generating a first set of transmission sub-states of a first particle and a second set of transmission sub-states of a second particle, the transmission sub-states representing a greater degree of correlations than the predetermined degree; transmitting the first set of transmission sub-states on respective ones of a first plurality of independent transmission channels; transmitting the second set of transmission sub-states on respective ones of a second plurality of independent transmission channels; combining the first set of transmission sub-states to generate receiver states of the first particle on respective ones of a first set of output channels; combining the second set of transmission sub-states to generate receiver states of the second particle on respective ones of a second set of output channels; determining whether or not to use or discard the states of the first and second particles depending on the output channels in which they arrive, wherein the states of the first and second particles which are determined to be used are available on corresponding channels of the first and second sets and are correlated to said predetermined degree.
Another aspect provides a system for generating particle states correlated to a predetermined degree at separated locations in a noisy environment, the system comprising: a source configured to generate a first set of transmission substates of a first particle and a second set of transmission sub-states of a second particle, the transmission substates representing a greater degree of correlation than the predetermined degree; a first plurality of transmission paths arranged between the source and a first decoder for conveying respectively the first set of transmission sub-states; a second plurality of transmission paths arranged between the source and a second decoder for conveying respectively the second set of transmission sub-states; wherein the first decoder is operable to combine the transmission sub-states to generate receiver states of the first particle on respective ones of a first set of output channels; wherein the second decoder is operable to combine the transmission substates to generate receiver states of the second particle on respective ones of a second set of output channels; and means for determining whether or not to use or discard the states of the first and second particles depending on the output channels in which they arrive, wherein the states of the first and second particles which are determined to be used are available on corresponding channels of the first and second sets and are correlated to said predetermined degree.
By way of explanation, consider the case where first and second receivers are located to receive signals from the first and second decoders respectively. The correlated states will appear on corresponding channels. That is, if the first receiver uses channels 1 and 3 then the second receiver should use 1 and 3. If the first receiver uses channels 2 and 4 then the second receiver should use 2 and 4. However, the receivers do not need to positively detect at this point which channels the particles appeared in. Indeed, doing the detection at this point in such a way that the state is not disturbed may in fact be difficult technologically. Therefore, the determining means may operate to determine if the particles are in channels 1 and 3, for example, without disturbing them if they are there, by measuring if any particle appears in any of the channels 2 or 4 on either side. If they do not appear in channels 2 or 4, they are in channels 1 and 3 and the protocol can continue. If they do appear in channels 2 or 4, the protocol is aborted (because the state has been disturbed by the measurement).
The second possibility is for a positive detecting step to occur a long way down the line. Thus in practice it could be anticipated that the receivers may well use all four channels on each side blindly (in a four channel system), that is performing operations as if there were a particle on the channel but without knowing (or needing to know) whether there actually is a particle in the channel or not. At the very end of the processing, users can detect where the particles are and use the results only if they came through coincident channels with correlated particles (i.e. 1 and 3 in both cases, or 2 and 4 in both cases).
It will be appreciated that throughout the specification “particles” are discussed. When a single particle is mentioned it is clear that it is not possible to guarantee that in practice the source produces exactly one particle because a source might not be perfect. Moreover, when linear transformations are referred to, it cannot be guaranteed in the real world that each piece of hardware performs a perfect linear transformation. However, the fact that the mathematical perfections do not find themselves realised in the real world does not detract from the fundamental usefulness of the systems and method discussed herein.
It will be appreciated that correlation between states can be provided with a number of factors, including correlation in time and/or polarisation. Both of these parameters can be used to generate more correlated states for the same number of originating particles.
We note that there is a correspondence between protocols for correcting errors during communication and protocols for distribution of entanglement. Therefore our protocols for error filtration during communication can be mapped in a straightforward way into protocols for filtering errors in distribution of entangled states (and vice versa).
The following description shows how to realise error filtration for quantum communication. The difference between error correction and error filtration is the following. In error correction the aim is actively to correct the errors that occur during transmission so that the decoded signal is as close as possible to the emitted signal. In error filtration that part of the signal which is affected by noise is discarded with high probability. What remains is a signal of reduced intensity but with less noise. This produces a signal of better quality than if no error filtration was carried out.
The error filtration protocols discussed in the following have a number of key advantages:                1. they are performed without controlled multi-particle interactions;        2. they do not require any special physical system for their implementation;        3. they do not require any special input quantum or classical state;        4. they can be applied to both bosons (e.g. photons) and fermions (e.g. electrons, or holes in semiconductors), and can be used for communication of single particle quantum states, multi-particle quantum states (e.g. coherent states, squeezed states, number states), entangled quantum states and also classical wave signals        5. they do not require active detection or active correction of errors.        
For a better understanding of the present invention and to show how the same may be carried into effect, reference will now be made by way of example to the accompanying drawings.